Wait, why don't I just combine the two schemes, instead of trying to settle for one or the other?
Firstly, a gatherer kills the deer and brings home the antler. Secondly, a builder carves the coins. Thirdly, the government mint stains the coins with a special pigment. Alternatively thirdly, the builder from secondly does the staining themselves, but they need to purchase the pigment from the government. The former version gives the government the option to simply confiscate a portion of the money if there's an antler glut. Those coins are then put in storage and circulated little by little once the glut is over, thus avoiding short-term currency fluctuations. In the latter version, this happens almost automatically, because they'll only be producing the pigment at the rate at which it is usually being bought. During the hypothetical glut, the pigment simply sells out, from which point on the gatherers and builders can either store the antler themselves until pigment becomes once again available, or can decide to use the antler to make other stuff.
The only problem with the otherwise very elegant latter version is that people would simply keep making as many high-denomination coins as there is antler available, using the extra pigment they can save by not making any cents at all. Thus, there'd still be inflation, and at the same time the denomination balance gets thrown off. That could be fixed by staining each denomination with a different pigment, though, unless I'm overlooking something.
As an added benefit, we now have three types of coins and three parties involved, so the divvying-up could be very straightforward indeed - one party gets the coronet, one party the cusps, one party the cents. At first glance, though, I'm not sure if there's a way to do that which is fair to everyone involved. Needs a bit more thought.
On the whole, this feels almost perfect, though!
Soon the Cold One took flight, yielded Goddess and field to the victor: The Lord of the Light.